LCAT Math, Geometry and Trigonometry
Circles for the LCAT: lesson and practice
Arc length and sector area are fractions of the whole circle: multiply the circumference or area by angle ÷ 360°. The equation (x − h)² + (y − k)² = r² has center (h, k) and radius r; the number on the right is r², not r. In radians, arc length = rθ.
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No calculator on the LCAT. LUMS’s official guidelines say calculators are not allowed, so every practice question here can be solved by hand.
| Find | Formula |
|---|---|
| Arc length (degrees) | (θ/360) × 2πr |
| Sector area | (θ/360) × πr² |
| Arc length (radians) | rθ |
| Center and radius | (x + 3)² + (y − 2)² = 25 → (−3, 2), r = 5 |
Completing the square
x² + 12x + y² − 6y + 41 = 0
(x + 6)² + (y − 3)² = 36 + 9 − 41 = 4
radius 2
(x + 6)² + (y − 3)² = 36 + 9 − 41 = 4
radius 2
Common mistakes
Center signsFlip the signs inside the brackets.
Using r² as rTake the square root of the right side.
Arc vs sectorArcs use circumference; sectors use area.
Sample questions with answers
1. Radius 10, central angle 120°. Arc length?
- 20π/3 (correct)
- 40π/3
- 100π/3
- 20π
(120/360) × 20π = 20π/3.
2. Center of (x − 8)² + (y − 3)² = 9?
- (−8, −3)
- (8, 3) (correct)
- (8, −3)
- (−8, 3)
Flip the signs: (8, 3).
3. Radius of (x − 1)² + (y + 4)² = 36?
- 36
- 18
- 6 (correct)
- 4
r² = 36, so r = 6.
Frequently asked questions
How do I find the radius from a circle equation?
Take the square root of the number on the right side of (x − h)² + (y − k)² = r².
What is arc length in radians?
Arc length = radius × angle in radians.
Is circles tested on the LCAT?
Yes. It falls under Geometry and Trigonometry, one of the content domains LUMS lists for the LCAT in its official sample questions.
Can I practice this without a calculator?
Yes. Every practice question here can be solved by hand, because calculators are not allowed on the LCAT.